Measurement of information is crucial to cybersecurity. One of these measures is distance measure between two quantum states.

• Static measures quantify how close two quantum states are [NC10, p. 399].
• Dynamic measures quantify how well information is preserved during a dynamic process [NC10, p. 399]. Dynamic measures can be derived from static measures.

Distance measures are defined in a way that makes sense to the analysis they are applied to, hence more than one distance measure exist in the literature, but two of these measures are in particularly wide use, namely trace distance and fidelity.

The focus here is trace distance, which for probability density functions $p_x,q_x$ and index set $x$ is defined to be [NC10, Eq. (9.1)]:

$D(p_x, q_x) \triangleq \frac{1}{2}\sum_x |p_x - q_x|.$

Trace distance is also called $L_1$ distance and Kolmogorov distance.

Extending the earlier definition to quantum states, the trace distance between density matrices $\rho$ and $\sigma$ is defined as [MM12, Sec. 3.11]:

$D(\rho,\sigma)\triangleq\frac{1}{2}\text{tr}|\rho-\sigma|.$

If